Lipschitz Free Spaces and Subsets of Finite-Dimensional Spaces

Abstract

We consider two questions on the geometry of Lipschitz free $p$-spaces ${\mathcal{F}}_p$, where $0<p\le 1$, over subsets of finite-dimensional vector spaces. We solve an open problem and show that if $(\mathcal{M},\rho )$ is an infinite doubling metric space (e.g., an infinite subset of an Euclidean space), then ${\mathcal{F}}_p(\mathcal{M},{\rho }^{\alpha })\simeq {\ell }_p$ for every $\alpha \in (0,1)$ and $0<p\le 1$. An upper bound on the Banach-Mazur distance between the spaces ${\mathcal{F}}_p([0,1{]}^d,|\cdot {|}^{\alpha })$ and ${\ell }_p$ is given. Moreover, we tackle a question due to arXiv:2006.08018v1 [math.FA] and expound the role of $p$, $d$ for the Lipschitz constant of a canonical, locally coordinatewise affine retraction from $(K,|\cdot {|}_1)$, where $K={\bigcup }_{Q\in \mathcal{R}}Q$ is a union of a collection $\varnothing \ne \mathcal{R}\subseteq \{Rw+R[0,1{]}^d:w\in {\mathbb{Z}}^d\}$ of cubes in ${\mathbb{R}}^d$ with side length $R>0$, into the Lipschitz free $p$-space ${\mathcal{F}}_p(V,|\cdot {|}_1)$ over their vertices.